Fréchet topology

The topological structure (topology) of an $ F $- space (a space of type $ F $; cf. also Fréchet space), i.e. a completely metrizable topological vector space. The term was introduced by S. Banach in honour of M. Fréchet. Many authors, however, demand additionally local convexity. A complete topological vector space is an $ F $- space if and only if it has a countable basis of neighbourhoods of the origin. The topology of an $ F $- space $ X $ can be given by means of an $ F $- norm, i.e. a function $ x \mapsto \| x \| $ satisfying:

i) $ \| x \| \geq 0 $ and $ \| x \| = 0 $ if and only if $ x = 0 $;

ii) $ \| {x + y } \| \leq \| x \| + \| y \| $ for all $ x,y \in X $;

iii) for each scalar $ \lambda $, $ {\lim\limits } _ {\| x \| \rightarrow 0 } \| {\lambda x } \| = 0 $, and for each $ x \in X $, $ {\lim\limits } _ {| \lambda | \rightarrow 0 } \| {\lambda x } \| = 0 $. This means that the (complete) topology of $ X $ can be given by means of a distance of the form $ d ( x,y ) = \| {x - y } \| $. The completion of any metrizable topological vector space (cf. Completion) is an $ F $- space and, consequently, the topology of any metric vector space can be given by means of a translation-invariant distance. Without loss of generality it can be assumed that $ \| {\lambda x } \| $ depends only upon $ | \lambda | $ and that the function $ | \lambda | \mapsto \| {\lambda x } \| $ is non-decreasing for each $ x \in X $. If one relaxes condition i) so that $ \| x \| = 0 $ can hold for a non-zero $ x $, one obtains an $ F $- semi-norm. The topology of an arbitrary topological vector space can be given by means of a family of $ F $- semi-norms; consequently, every complete topological vector space is an inverse (projective) limit of a directed family of $ F $- spaces.

Important classes of $ F $-spaces.

Locally convex $ F $-spaces.

Such spaces are also called spaces of type $ B _ {o} $( some authors call them just Fréchet spaces, but see Fréchet space). The topology of such a space $ X $ can be given by means of an increasing sequence of (homogeneous) semi-norms

$$ \tag{a1 } \left \| x \right \| _ {1} \leq \left \| x \right \| _ {2} \leq \dots , \textrm{ for all } x \in X, $$

so that a sequence $ ( x _ {k} ) $ of elements of $ X $ tends to $ 0 $ if and only if $ {\lim\limits } _ {k} \| {x _ {k} } \| _ {n} = 0 $ for $ n = 1,2, \dots $. An $ F $- norm giving this topology can be written as

$$ \left \| x \right \| = \sum _ { 1 } ^ \infty 2 ^ {- n } { \frac{\left \| x \right \| _ {n} }{1 + \left \| x \right \| _ {n} } } . $$

If $ T $ is a continuous linear operator from a $ B _ {o} $- space $ X $ to a $ B _ {o} $- space $ Y $, then for each $ n $ there are a $ k ( n ) $ and a $ C _ {n} > 0 $ such that $ \| {Tx } \| _ {n} ^ {( Y ) } \leq C _ {n} \| x \| _ {k ( n ) } ^ {( X ) } $, $ x \in X $, for all $ n $( it is important here that the systems of semi-norms giving, respectively, the topologies of $ X $ and $ Y $ satisfy (a1)). The dual space $ X ^ \prime $ of a $ B _ {o} $- space $ X $( the space of all continuous linear functionals provided with the topology of uniform convergence on bounded sets) is said to be an $ LF $- space; it is non-metrizable (unless $ X $ is a Banach space). Any space of type $ B _ {o} $ is an inverse (projective) limit of a sequence of Banach spaces.

Complete locally bounded spaces.

A topological vector space $ X $ is said to be locally bounded if it has a bounded neighbourhood of the origin (then it has a basis of such neighbourhoods consisting of bounded sets). The topology of such a space $ X $ is metrizable and can be given by means of a $ p $- homogeneous norm, $ 0 < p \leq 1 $, i.e. an $ F $- norm satisfying instead of iii) the more restrictive condition

iiia) $ \| {\lambda x } \| = | \lambda | ^ {p} \| x \| $ for all scalars $ \lambda $ and all $ x \in X $.

For this reason, locally bounded spaces are sometimes called $ p $- normed spaces. The class of all Banach spaces is exactly the intersection of the class of locally convex $ F $- spaces and the class of complete locally bounded spaces. The dual space of a locally bounded space can be trivial, i.e. consist only of a zero functional.

Locally pseudo-convex $ F $-spaces.

They are like $ B _ {o} $- spaces, but with $ p $- homogeneous semi-norms instead of homogeneous semi-norms (the exponent $ p $ may depend upon the semi-norm). This class contains the class of locally convex $ F $- spaces and the class of complete locally bounded spaces.

Examples of $ F $-spaces.

The space $ S [ 0,1 ] $ of all Lebesgue-measurable functions on the unit interval with the topology of convergence in measure (asymptotic convergence) is a space of type $ F $. Its topology can be given by the $ F $- norm

$$ \left \| x \right \| = \int\limits _ { 0 } ^ { 1 } { { \frac{\left | {x ( t ) } \right | }{1 + \left | {x ( t ) } \right | } } } {dt } . $$

This space is not locally pseudo-convex.

The space $ C ^ \infty [ 0,1 ] $ of all infinitely differentiable functions on the unit interval with the topology of unform convergence of functions together with all derivatives is a $ B _ {o} $- space. Its topology can be given by semi-norms

$$ \left \| x \right \| _ {n} = $$

$$ = \max \left \{ \max _ {[ 0,1 ] } \left | {x ( t ) } \right | , \max _ {[ 0,1 ] } \left | {x ^ \prime ( t ) } \right | \dots \max _ {[ 0,1 ] } \left | {x ^ {( n - 1 ) } ( t ) } \right | \right \} . $$

The space $ {\mathcal E} $ of all entire functions of one complex variable with the topology of uniform convergence on compact subsets of the complex plane is a $ B _ {o} $- space. Its topology can be given by the semi-norms

$$ \left \| x \right \| _ {n} = \max _ {\left | \zeta \right | \leq n } \left | {x ( \zeta ) } \right | . $$

The space $ L _ {p} [ 0,1 ] $ on the unit interval, $ 0 < p < 1 $, is a complete locally bounded space with trivial dual. Its topology can be given by $ \| x \| _ {p} = \int _ {0} ^ {1} {| {x ( t ) } | ^ {p} } {dt } $( its discrete analogue, the space $ l _ {p} $ of all sequences summable with the $ p $- th power, has a non-trivial dual).

The space $ L ^ {0 + } [ 0,1 ] = \cap _ {0 < p \leq 1 } L _ {p} [ 0,1 ] $ with the semi-norms $ \| x \| _ {p _ {n} } $, where $ 0 < p _ {n} \rightarrow 0 $, is a locally pseudo-convex space of type $ F $ which is not locally bounded.

General facts about $ F $-spaces.

A linear operator between $ F $ spaces is continuous if and only if it maps bounded sets onto bounded sets.

Let $ {\mathcal A} $ be a family of continuous linear operators from an $ F $- space $ X $ to an $ F $- space space $ Y $. If the set $ \{ {Tx } : {T \in {\mathcal A} } \} $ is bounded in $ Y $ for each fixed $ x \in X $, then $ {\mathcal A} $ is equicontinuous (the Mazur–Orlicz theorem; it is a theorem of Banach–Steinhaus type, cf. also Banach–Steinhaus theorem).

If $ X $ and $ Y $ are $ F $- spaces and $ ( T _ {n} ) $ is a sequence of continuous linear operators from $ X $ to $ Y $ such that for each $ x \in X $ the limit $ Tx = {\lim\limits } _ {n} T _ {n} x $ exists, then $ x \mapsto Tx $ is a continuous linear operator from $ X $ to $ Y $.

The image of an open set under a continuous linear operator between $ F $- spaces is open (the open mapping theorem).

The graph of a linear operator $ T $ between $ F $- spaces is closed if and only if $ T $ is continuous (the closed graph theorem).

If a one-to-one continuous linear operator maps an $ F $- space onto an $ F $- space, then the inverse operator is continuous (the inverse operator theorem).

A separately continuous bilinear mapping between $ F $- spaces is jointly continuous (cf. also Continuous function).

References

[a1]	S. Banach, "Théorie des operations lineaires" , Warszawa (1932)
[a2]	N. Bourbaki, "Espaces vectorielles topologiques" , Paris (1981) pp. Chapt. 1–5
[a3]	N. Dunford, J.T. Schwartz, "Linear operators" , I. General theory , Wiley, reprint (1988)
[a4]	A. Grothendieck, "Topological vector spaces" , New York (1973)
[a5]	H. Jarchow, "Locally convex spaces" , Teubner (1981)
[a6]	G. Köthe, "Topological vector spaces" , I–II , New York (1969–1979)
[a7]	S. Rolewicz, "Metric linear spaces" , PWN & Reidel (1972)
[a8]	H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[a9]	L. Waelbroeck, "Topological vector spaces and algebras" , Lecture Notes in Mathematics , 230 , Springer (1971)
[a10]	A. Wilansky, "Modern methods in topological vector spaces" , New York (1978)